Spohn H. Dynamics of Charged Particles and their Radiation Field

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2.3 Coupled Maxwell’s and Newton’s equations 13

The (E, B) ﬁelds in (2.26) are not completely arbitrary. They are subject to the

Maxwell equations with source (ρ, j).Inother words, we have divided all charges

into a single charged particle whose motion is determined through (2.26) and the

rest whose motion is taken to be known.

The Newton–Lorentz equations (2.26) are of Hamiltonian form. To see this we

introduce vector potentials φ, A such that

E(x, t) =−∇φ(x, t) − c

−1

∂

A(x, t), B(x, t) =∇×A(x, t). (2.27)

Then the Lagrangian associated with (2.26) is

L(q, ˙q, t) =−m

(1 − c

−2

˙q

)

1/2

− e(φ (q, t ) − c

−1

˙q · A(q, t)) . (2.28)

To switch to the Hamiltonian framework, one introduces the canonical momentum

p = m

γ(˙q) ˙q +

A(q, t) (2.29)

and obtains the Hamiltonian function

H(q, p, t) =



(c p− e A(q, t))

+ m



1/2

+ eφ(q, t). (2.30)

In particular, whenever the ﬁelds are time independent, the energy

E(q, v) = m

γ(v) + eφ(q) (2.31)

is conserved along the solution trajectories of (2.26).

It should be noted that in general the solutions to Newton’s equations of motion

(2.26) will have a complicated structure even for time-independent ﬁelds. This

has been amply demonstrated for particular cases. Depending on how the external

ﬁelds are chosen, the motion would range from regular to fully chaotic with a

mixed phase space as a rule.

2.3 Coupled Maxwell’s and Newton’s equations

While for most practical purposes, barring a few exceptional cases, it sufﬁces to

use either Maxwell’s equations with prescribed sources or Newton’s equations

with prescribed forces, from a more fundamental point of view such a procedure is

unsatisfactory. Physically it would seem more natural to have a coupled system of

equations for the time evolution of the charged particles together with their elec-

tromagnetic ﬁeld and to regard the two cases discussed above as emerging limit

situations. If for the moment we restrict ourselves to a single particle, it is obvious

14 Acharge coupled to its electromagnetic ﬁeld

how to proceed. From (2.2), (2.3) we have

∂

B(x, t ) =−∇×E(x, t ),

∂

E(x, t) =∇×B(x, t) − eδ(x − q(t))v(t) (2.32)

with the constraints

∇·E(x, t ) = eδ(x − q(t)) , ∇·B(x, t) = 0 . (2.33)

Moreover, from (2.26) we have



γ v(t)



= e



(q(t)) + E(q(t), t) + v(t) × (B

(q(t)) + B(q(t), t))



(2.34)

We added the external electromagnetic ﬁelds E

, B

, which will play a promi-

nent role later on. They are derived from potentials as

=−∇φ

, B

=∇×A

. (2.35)

We assume the potentials to be time independent for simplicity, although a con-

siderable part of the theory to be developed will work also for time-dependent

ﬁelds. As before, (2.32)–(2.34) are to be solved as an initial value problem. Thus

E(x, 0), B(x, 0), q(0), and v(0) are supposed to be given. Note that the continu-

ity equation is satisﬁed by ﬁat.

Equations (2.32), (2.34) are the stationary points of a Lagrangian action, which

strengthens our trust in these equations, since every microscopic classical evolution

equation seems to be of that form. We continue to use the underlying electromag-

netic potentials as in (2.27), (2.35). Then the action for (2.32), (2.34) reads

A([q,φ,A]) =





− m

(1 −˙q(t)

)

1/2

− e



(q(t)) + φ(q(t), t)

−˙q(t) · ( A

(q(t)) + A(q(t), t))







(∇φ(x, t) + ∂

A(x, t))

− (∇×A(x, t))



(2.36)

The only difﬁculty is that (2.32) and (2.33) taken together with (2.34) make no

proper mathematical sense. As explained, the solution of the Maxwell equations is

singular at x = q(t), and in the Lorentz force we are asked to evaluate the ﬁelds

precisely at that point. One might be tempted to put the blame on the mathematics

which refuses to handle equations as singular as (2.32)–(2.34). However before

such a drastic conclusion is drawn, the physics should be properly understood. The

point charge carries along with it a potential which at short distances diverges as

2.3 Coupled Maxwell’s and Newton’s equations 15

the Coulomb potential, cf. (2.24), and which therefore has the electrostatic energy



{|x−q(t)|≤R}

x E(x, t )





drr

−2

)



drr

−2

=∞. (2.37)

Taken literally, such an object would have an inﬁnite mass and hence would not

respond to external forces. It would keep its velocity for ever, which is inconsistent

with what is observed.

Thus we are forced to regularize at short distances the coupled system consisting

of the Maxwell equations and Newton’s equation of motion with the Lorentz force.

In carrying out such a program there are two, in part, complementary points of

view. The ﬁrst one, which we will not follow here, starts from the idea that regu-

larization is a mathematical device with the sole purpose of making sense of a sin-

gular mathematical object through a suitable limiting procedure. To illustrate this

approach we can think of the following prominent mathematical physics example.

The free scalar ﬁeld, φ(x),inEuclidean quantum ﬁeld theory in 1 + 1 dimensions

ﬂuctuates so wildly at short distances that an interaction such as



xV(φ(x))

with V (φ) = φ

+ λφ

cannot be properly deﬁned. One way, not necessarily op-

timal, to regularize the theory is to introduce a spatial lattice with spacing a.

Such a lattice ﬁeld theory is well deﬁned in any ﬁnite volume. On taking the

limit a → 0 along with a simultaneous readjustment of the interaction potential,

V (φ) = V

(φ),aEuclidean-invariant, interacting quantum ﬁeld theory is obtained.

Ideally this limit theory should be independent of the regularization scheme. For

instance one could start with the free scalar ﬁeld in the continuum and regularize

φ(x) as φ ∗ g(x) with a suitable test function g concentrated at 0. Then the reg-

ularized interaction is



xV(φ ∗ g(x)) and in the limit g(y) → δ(y) a quantum

ﬁeld theory should be obtained identical to the one from the lattice regularization.

In the second approach one argues that there is a physical cutoff coming from a

more reﬁned theory, which is then modeled in a phenomenological way. While this

is a standard procedure, it is worthwhile to illustrate it again with a concrete exam-

ple. Consider a large number (

∼

) of He

atoms in a container of adjustable

size and suppose we are interested in computing their free energy according to

the rules of statistical mechanics. The more reﬁned theory is here nonrelativistic

quantum mechanics which treats the electrons and nuclei as point particles carry-

ing a spin

, respectively spin 0. As far as we can tell, this model approximately

covers the temperature range T = 0KtoT = 10

K, i.e. way beyond dissocia-

tion, and the density range ρ = 0toρ = ρ

, the density of close packing. Beyond

these limits relativistic effects must be taken into account. However, there is a

more limited range where we can get away with a model of classical point par-

ticles interacting through an effective potential of Lennard-Jones type. Once this

16 Acharge coupled to its electromagnetic ﬁeld

pair potential is speciﬁed classical statistical mechanics makes well-deﬁned pre-

dictions at any T ,ρ. There is no limitation in principle. Only outside a certain

range of parameters would the classical model lose the correspondence with the

real world. Already from the way the physical cutoff is described, there is a con-

siderable amount of vagueness. How much error should we allow in the free en-

ergy? What about more detailed properties like density correlations? An effective

potential can be deﬁned quantum mechanically, but it is temperature dependent

and never strictly a pair potential. Despite all these imprecisions and shortcom-

ings, the equilibrium theory of ﬂuids relies heavily on the availability of a classical

model.

In the same spirit we modify the coupled Maxwell and Newton equations by

introducing an extended charge distribution as a phenomenological model for the

omitted quantum electrodynamics. The charge distribution is stabilized by strong

interactions which act outside the realm of electromagnetic forces. On the classical

level, say, an electron appears as an extended charged object with a size roughly of

the order of its Compton wavelength, i.e. 4 × 10

−11

cm. We impose the obvious

condition that the extended charge distribution has to be adjusted such that, in

the range where classical electrodynamics is applicable, the coupled Maxwell and

Newton equations correctly reproduce the empirical observations.

Such general clauses seem to leave a lot of freedom in the construction of the

theory. However, charge conservation and the Lagrangian form of the equations

of motion severely limit the possibilities. In fact, essentially only two models of

extended charge distribution have been investigated so far.

(i) The semirelativistic Abraham model of a rigid charge distribution. The

charge e is assumed to be smeared out over a ball of radius R

. This means that in

(2.32)–(2.34) the δ-function is replaced by a smooth charge distribution eϕ. ϕ(x)

is taken to be radial, vanishing for |x| > R

, and normalized as



xϕ(x) = 1.

Equivalently, having (2.32)–(2.34) recast in Fourier space, the couplings between

the ﬁeld modes with |k|

 1/R

and the particle become suppressed. This partic-

ular choice for the internal structure of the charge is called the Abraham model

(for a single nonrotating charge). For zero coupling the model is relativistic. How-

ever, ϕ is taken to be rigid, thus velocity independent in a prescribed coordinate

frame, which breaks Lorentz invariance. The standard examples are that the charge

is uniformly distributed either over the ball, ϕ(x) = (4π R

/3)

−1

for |x|≤R

ϕ(x) = 0 otherwise, or over the sphere, ϕ(x) = (4π R

)

−1

δ(|x|−R

).Inthe

quantized version of the Abraham model, cf. chapter 13 below, often a sharp cutoff

in Fourier space is adopted, i.e. ϕ(k) = (2π)

−3/2

for |k|≤ = R

−1

, ϕ(k) = 0

otherwise; this has the slight disadvantage of being oscillating and having slow

decay in position space.

2.4 The Abraham model 17

Once the charge distribution is extended, besides its center of charge, also ro-

tational degrees of freedom must be taken into account. The Abraham model al-

lowing for a spinning charge will be discussed in chapter 10. Since the dynamical

behavior then becomes more complex, it is advisable to omit spin in the ﬁrst round.

The Abraham model will be studied in considerable detail. While deﬁned for

all velocities |v(t)| < c,itbecomes empirically inaccurate at velocities close to c.

Despite this drawback we hope that the Abraham model will serve as a blueprint

towards a more realistic description of matter.

(ii) The Lorentz model of a relativistically rigid charge distribution. More in

accord with special relativity is to require that eϕ is the charge distribution in the

momentary rest frame of the particle. While such a principle was already stated by

Lorentz and Poincar

e, a satisfactory dynamical theory has been arrived at only very

recently. As we will explain in section 2.5, in a relativistic theory translational and

rotational degrees of freedom are intrinsically coupled. To gain an understanding

of how relativistic invariance would modify the theory, we insert some features of

the Lorentz model, although our understanding of its dynamical properties is far

less developed than that of the Abraham model.

We emphasize that for extended charge models the diameter R

of the charge

distribution deﬁnes a length (and upon dividing by c also a time) scale, relative to

which the approximate validity of effective theories, like the Lorentz–Dirac equa-

tion, can be addressed quantitatively. In fact, apart from the external forces, R

the only natural length scale available.

2.4 The Abraham model

Following Abraham, we model the charged particle as a spherically symmetric,

rigid body to which the charge elements are permanently attached. The charge dis-

tribution is prescribed and independent of the particle’s velocity, which singles out

the laboratory frame. In a relativistic theory the charge distribution would appear

to be Lorentz contracted. To be speciﬁc the charge distribution eϕ is assumed to

be smooth, radial, and supported in a ball of radius R

, and normalized to e, i.e.

Condition (C):

ϕ ∈ C

∞

), ϕ(x) = ϕ

(|x|), ϕ(x) = 0 for |x|≥R



xϕ(x) = 1 .

(2.38)

3 eϕ(x) is the charge distribution and ϕ(k) is the form factor, since in Fourier

space it multiplies the current as (2π)

3/2

ϕ(k)



j(k, t). 3

18 Acharge coupled to its electromagnetic ﬁeld

Our goal is to set up the Abraham model as a well-deﬁned dynamical system.

Usually this point is taken for granted. Since the occurrence of ill-deﬁned equa-

tions of motion was one of our main objections to the δ-charge, it is worthwhile to

understand why this objection is no longer valid for a smeared out δ.

The equations of motion for the Abraham model are

∂

B(x, t ) =−∇×E(x, t ),

∂

E(x, t) =∇×B(x, t) − eϕ(x − q(t))v(t), (2.39)

∇·E(x, t ) = eϕ(x − q(t)) , ∇·B(x, t) = 0 , (2.40)



γ v(t)



= e



(q(t)) + E

(q(t), t ) + v(t) × (B

(q(t)) + B

(q(t), t ))



(2.41)

where we have set c = 1. In (2.41) we use the shorthand E

(x) = E ∗ ϕ(x) and

(x) = B ∗ ϕ(x) so as to resemble (2.34). Strictly speaking also E

, B

should

be smeared over ϕ;however, this would only amount to a redeﬁnition of the exter-

nal potentials. In contrast to Newton’s equations of motion (2.26), for the Abraham

model we denote the mechanical mass of the particle by m

to emphasize that this

bare mass will differ from the observed mass of the compound object “particle

plus surrounding Coulomb ﬁeld”. The external potentials φ

, A

can be fairly

arbitrary. We only require them and their derivatives to be smooth and locally

bounded, to avoid too strong local oscillations. No condition on the increase at in-

ﬁnity is needed, since |v(t)|≤1. However, it is convenient to have the energy, as

deﬁned in (2.44), uniformly bounded from below. To keep things simple we make

the (unnecessarily strong) assumptions

Condition (P):

∈ C

∞

), A

∈ C

∞

, R

), φ

≥

φ>−∞ . (2.42)

Moreover, there exists a constant C such that |∇φ

|≤C, |∇ A

|≤C.

The Abraham model is derived from the Lagrangian

L =−m

(1 −˙q

)

1/2

− e



(q) + φ

(q) −˙q · A

(q)







(∇φ + ∂

− (∇×A)



. (2.43)

Correspondingly, the energy

E(E, B, q, v) = m

γ(v) + eφ

(q) +





E(x)

+ B(x)



(2.44)

is conserved.

2.4 The Abraham model 19

As for any dynamical system, the ﬁrst step in dealing with (2.39)–

(2.41) is to construct a suitable phase space. The dynamical variables are

(E(x), B(x), q, v) = Y which is called a state of the system. We have q ∈

, v ∈

V ={v ||v| < 1}.Inaddition, the energy (2.44) should be bounded. Thus it is nat-

ural to introduce the (real) Hilbert space

= L

, R

) (2.45)

with norm E=(



x|E(x)|

)

1/2

and to deﬁne L as the set of states satisfying

Y 

=E+B+|q|+|γ(v)v| < ∞ . (2.46)

In particular for the ﬁeld energy,

(E

+B

)<∞. The norm || · ||

gives

rise to the metric

d(Y

, Y

) =E

− E

+B

− B

+|q

− q

|+|γ(v

− γ(v

| .

(2.47)

In addition, the constraints (2.40) have to be satisﬁed. Thus the phase space,

for the Abraham model is the nonlinear submanifold of

L deﬁned through

∇·E(x) = eϕ(x − q), ∇·B(x) = 0 . (2.48)

M inherits its metric from L.

On various occasions below we will need the property that the system forgets

its initial ﬁeld data. For this purpose it is helpful to have a little bit of smoothness

and some decay at inﬁnity. Formally we introduce the “good” subset

⊂ M,

0 ≤ σ ≤ 1, consisting of ﬁelds such that componentwise and outside a ball of

radius R

, |x|≥R

,wehave

|E(x)|+|B(x)|+|x|(|∇ E(x)|+|∇B(x)|) ≤ C |x|

−1−σ

. (2.49)

The Li

enard–Wiechert ﬁelds (2.24), (2.25) are included in

; moreover, M

dense in

M.However M

=∅for σ>1, by Gauss’s law (2.40) with e = 0.

The evolution equations (2.39)–(2.41) are of the general form

Y (t) = F(Y (t)) (2.50)

with Y (0) = Y

∈ M.Weturn to the question of the existence and uniqueness of

solutions of the Abraham model (2.50).

Theorem 2.1 (Existence of the dynamics for the Abraham model). Let the con-

ditions (C) and (P) hold and let Y

= (E

(x), B

(x), q

, v

) ∈ M. Then the

20 Acharge coupled to its electromagnetic ﬁeld

integral equation associated with (2.50),

Y (t) = Y



dsF(Y (s)) , (2.51)

has a unique solution Y (t) = (E(x, t), B(x, t), q(t), v(t)) ∈

M, which is contin-

uous in t and satisﬁes Y (0) = Y

. Along the solution trajectory

E(Y (t)) = E (Y

) (2.52)

for all t, i.e. the energy is conserved.

For short times existence and uniqueness follow through the contraction mapping

principle with constants depending only on the initial energy. For smooth initial

data, energy conservation is veriﬁed directly and by continuity it extends to all

ﬁnite-energy data. Thus we can construct iteratively the solution for all times.

We ﬁrst summarize some properties of the Maxwell equations. They follow di-

rectly from the Fourier and convolution representations (2.12), (2.13), respectively

(2.16), (2.17).

Lemma 2.2 In the Maxwell equations (2.2), (2.3), let eϕ(x, t) = eϕ(x −

q(t)), j(x, t) = eϕ(x − q(t))v(t), with prescribed t → (q(t), v(t)) continuous.

Then (2.2), (2.3) has a unique solution in C(

R, L

⊕ L

). The solution map

, B

) → (E(t), B(t)) depends continuously on (q(t), v(t)).

Proof of Theorem 2.1: Let b > 0beﬁxed and choose initial data such that

E(Y

) ≤ b .

(i) There exists a unique solution Y (t) ∈ C([0,δ],

M) for δ = δ(b) sufﬁciently

small.

We write (2.41) in the form



γ v(t)



= F

(t) + F

ini

(t) + F

self

(t) (2.53)

by inserting E(x, t), B(x, t) from the Maxwell equations according to (2.16),

(2.17). Let

(x) = e



k|ϕ(k)|

ik·x

|k|

sin |k|t

= (2π)





ϕ(y)ϕ(y



)

4πt

δ(|y + x − y



|−t). (2.54)

2.4 The Abraham model 21

Then

(t) = e

−



(q(t)) + v(t) × B

(q(t))



, (2.55)

ini

(t) =



xeϕ(x − q(t))



∂

∗ E

(x) +∇×G

∗ B

(x)

+ v(t) × ∂

∗ B

(x) − v(t) × (∇×G

∗ E

(x))



, (2.56)

self

(t) =





−∇W

t−s

(q(t) − q(s)) − v(s)∂

t−s

(q(t) − q(s))

+ v(t) × (∇×v(s)W

t−s

(q(t) − q(s)))



. (2.57)

We now integrate both sides of (2.53) over the time interval [0, t]. The resulting

expression is regarded as a map from the trajectory t → (q(t), v(t)),0≤ t ≤ δ,

to the trajectory t → ( ¯q(t), ¯v(t)) and is deﬁned by

¯q(t) = q



ds v(s), (2.58)

γ(¯v(t)) ¯v(t) = m

γ(v





(s) + F

ini

(s) + F

self

(s)



where F

(s), F

ini

(s), and F

self

(s) are functionals of q(·), v(·) according to

(2.55)–(2.57). Since ϕ, W,φ

, and A

are smooth, this map is a contraction in

C([0, t ],

× V), i.e.

sup

0≤s≤t



| ¯q

(s) − ¯q

(s)|+|¯v

(s) − ¯v

(s)|



≤ c(t, b) sup

0≤s≤t



(s) − q

(s)|+|v

(s) − v

(s)|



, (2.59)

with a constant c(t, b) depending on b and c(t, b)<1 for sufﬁciently small t . Such

a map has a unique ﬁxed point which is the desired solution (q(t), v(t)).Bythe

Maxwell equations also B(x, t), E(x, t) are uniquely determined.

(ii) The solution map Y

→ Y (t) is continuous in M.

This follows from Lemma 2.2 and the continuous dependence of (q(t), v(t)) on

the initial data.

(iii) The energy is conserved.

We choose smooth initial ﬁelds such that E, B ∈ C

∞

) and

|∇

E(x)|+|∇

B(x)|≤C(1 +|x|)

−(2+|α|)

. (2.60)

22 Acharge coupled to its electromagnetic ﬁeld

Here α = (α

,α

) is a multi-index with α

= 0, 1, 2,.... This subset is dense

M.Bythe convolution representation (2.16), (2.17) of the solution to the

Maxwell equations we have E(x, t ), B(x, t) ∈ C

([0,δ] × R

) and |E(x, t)|+

|B(x, t)|≤C(1 +|x|)

−2

. Also v(t) ∈ C

([0,δ]). Thus we are allowed to differ-

entiate,

E(Y (t)) = γ

v ·˙v + v ·∇φ

(q) +



x(E · ∂

E + B · ∂





E · (∇×B) − B · (∇×E)



= 0 , (2.61)

since the ﬁelds decay and hence the surface terms vanish. Thus

E(Y (t)) = E (Y

)

for 0 ≤ t ≤ δ.Bycontinuity this equality extends to all of

(iv) The global solution exists.

From (iii) we know that

E(Y (δ)) = E (Y

) ≤ b. Thus we can repeat the previous

argument for δ ≤ t ≤ 2δ, etc. Backwards in time we still have the solution (2.16),

(2.17) of the Maxwell equations, only the retarded ﬁelds have to be replaced by

the advanced ﬁelds. Thereby we obtain the solution for all times.

Theorem 2.1 ensures the existence and uniqueness of solutions for the Abraham

model. For initial data Y

∈ M the solution trajectory t → Y (t) lies in the phase

space

M,iscontinuous in t, and its energy is conserved. We have thus established

the basis for further investigations on the dynamics of the Abraham model.

2.5 The relativistically covariant Lorentz model

To improve on the semirelativistic Abraham model, following Lorentz, it is nat-

ural to assume that when viewed in a momentary inertial rest frame the charge

and mass distribution of the particle remain unchanged. This is what one would

call a relativistically rigid extended charge. Our requirement ﬁxes uniquely the

four-current density. The equations of motion then follow from a relativistically

covariant action.

Forobvious reasons we will switch to relativistic notation, where we follow the

conventions of Misner, Thorne, and Wheeler. Our arena is the Minkowski space-

time

.ALorentz frame, F

,inM

is speciﬁed through the tetrad {e

, e

}

of ﬁxed unit vectors. They have the inner product

· e

= g

µν

, (2.62)

where g

µν

is the metric tensor with g

=−1, g

µµ

= 1, µ = 1, 2, 3, and g

µν

= 0

otherwise. Therefore

can be identiﬁed with R

1,3

.Inthe given basis, a vector

x ∈

is expanded as

x = x

(2.63)